Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

a set $T = {Simple, Limit,Colimit,Power}$ of four type symbols; the meaning of this set will become clear in the following definition. Finally, given a set $Y$ , we need the set $Dia(Y /E)$ of finite diagrams $D : D --> E$ , where $D$ is a diagram scheme (a quiver, see (Mazzola, 2002, C.2.2)), the vertexes $d,e,f,...$ being elements of $Y$ , and where for each pair $e,f$ or vertexes, the arrows $i : e --> f$ are identified by positive natural numbers $i = 1,2,3,...$ , i.e., an arrow is a triple $(e,f,i)$ . Finally, we denote by $* Dia (Y/E)$ the disjoint union $E | ~| Dia(Y/E)$ . By abuse of language, we call the elements of $E$ , when embedded in $* Dia (Y /E)$ , trivial diagrams (in fact, in previous definitions of form semiotics, we used a special denotator to grasp trivial diagrams, and the present formalism is just a more direct restatement of that artifical setup).

Definition 1 Given the type set $T$ , a topos $E$ , an address subcategory $R$ , a set $D$ , the elements of which are called denotators, and a set $F$ , the elements of which are called forms, a form semiotic is the data of a type map $T : F --> T$ , a diagram map $* D : F --> Dia (F/E)$ , an identifier map $Id : F --> E$ , a coordinate map $C : D --> E$ , a denotator name map $DN : D --> D$ , form name map $FN : F --> D$ , and a denotator form map $DF : D --> F$ . These data are required to have the following properties:

A form $F$ is uniquely determined by its name $fn = FN (F)$ , its identifier $id = Id(F)$ , its type $t = T(F)$ , and its diagram $d = D(F )$ . We therefore also denote a form $F$ by the DenoteX¹

¹
DenoteX is an ASCII-based denotation language for denotators. An EBNF specification of DenoteX is available from http://www.ifi.unizh.ch/mml/musicmedia/downloads.php4.
symbol $fn : id.t(d)$ and also write $F ~ fn : id.t(d)$ to indicate that $F$ is determined by its four images.
A form’s identifier $Id(F )$ is supposed to be in $M ono(E)$ ; we call the domain $dom(Id(F ))$ the form’s space and denote it by $space(F )$ , whereas the codomain $codom(Id(F ))$ is called the form’s frame (space) and is denoted by $f rame(F)$ .
The domain $dom(C(D))$ of the coordinate $C(D)$ of a denotator $D$ is supposed to be an address $a = A(D)$ within $R$ ; $a$ is called the denotator’s address.
The coordinate codomain $codom(C(D))$ of a denotator $D$ is the space of the denotator’s form $F (D)$ . The image $Id(F (D))o C(D)$ is also called the frame coordinate of $D$ .
A denotator $D$ is uniquely determined by its name $dn = DN (D)$ , its form $f = DF (D)$ , and its coordinate $c = C(D)$ , with address $a$ . We therefore also write down a denotator by its DenoteX symbol $dn : a@f (c)$ and also write $D ~ dn : a@f (c)$ to indicate that $D$ is determined by these data. Logically, the address is superfluous since it is already contained in the coordinate, the stress of this important information is however advantageous for an immediate recognition.
If a form’s type is simple, i.e., $T(F) = Simple$ , then its diagram $D(F)$ is an address $A$ . This address equals the frame space $frame(F )$ of $F$ .